Introduction
Advantages and Disadvantages
Algorithm
TIME COMPLEXITY
Splay Tree
Cheruku Ravi Teja
November 14, 2011
Cheruku Ravi Teja
Splay Tree
Introduction
Advantages and Disadvantages
Algorithm
TIME COMPLEXITY
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2
3
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Introduction
Real Time Applications
Advantages and Disadvantages
Algorithm
Operations
When to Splay
Splaying
Zig step
Zig-Zig step
Zig-Zag step
Results of Splaying
Insertion
Deletion
TIME COMPLEXITY
Cheruku Ravi Teja
Splay Tree
Introduction
Advantages and Disadvantages
Algorithm
TIME COMPLEXITY
Real Time Applications
Introduction
Splay trees are self branching binary search tree which has the
property of reaccessing the elements quickly that which are
recently accessed.
The performance of the Splay trees depends on the self
balancing and self optimizing.
The worst case with this splay tree algorithm is that this will
sequentially access all the elements of the tree which makes
tree unbalanced.
Cheruku Ravi Teja
Splay Tree
Introduction
Advantages and Disadvantages
Algorithm
TIME COMPLEXITY
Real Time Applications
Real Time Applications
It is used to implement caches.
It has the ability of not to store any data, which results in
minimization of memory requirements.
It can also be used for data compression, e.g.dynamic huffman
coding.
Cheruku Ravi Teja
Splay Tree
Introduction
Advantages and Disadvantages
Algorithm
TIME COMPLEXITY
Advantages
It is easy to implement than other self branching binary search
trees, such as Red black trees or AVL trees. Much simpler to
code than AVL, Red Black trees.
It requires less space as no balance information is required.
Cheruku Ravi Teja
Splay Tree
Introduction
Advantages and Disadvantages
Algorithm
TIME COMPLEXITY
Disadvantages
More local adjustments during Search operations.
Invidual operations can be expensive, drawback for real-time
applications.
The main disadvantage of Splay trees is the height. After
accessing all 0 n0 elements in the tree, the height of the tree
corresponds to worst case access time.
Cheruku Ravi Teja
Splay Tree
Introduction
Advantages and Disadvantages
Algorithm
TIME COMPLEXITY
Operations
When to Splay
Splaying
Zig step
Zig-Zig step
Zig-Zag step
Results of Splaying
Insertion
Deletion
Algorithm
Splay trees are self adjusting binary search trees which performs
basic operations such as
Search
Insertion
Deletion
Search,Insert,Delete operations are like in Binary Search trees,
except at the end of each operation, a special step called Splaying
is done.Splaying the tree rearranges the tree so that element is
placed at the root of the tree.It uses tree rotations to bring the
element to the top.
Cheruku Ravi Teja
Splay Tree
Introduction
Advantages and Disadvantages
Algorithm
TIME COMPLEXITY
Operations
When to Splay
Splaying
Zig step
Zig-Zig step
Zig-Zag step
Results of Splaying
Insertion
Deletion
Operations
The main basic operations of the Splay tree are
Search
Insert
Delete
Splaying
Cheruku Ravi Teja
Splay Tree
Introduction
Advantages and Disadvantages
Algorithm
TIME COMPLEXITY
Operations
When to Splay
Splaying
Zig step
Zig-Zig step
Zig-Zag step
Results of Splaying
Insertion
Deletion
When to Splay
Search: Splay node where key was found.
Insert: When an item is inserted , a Splay is performed. As a
result, the newly inserted node becomes the root of the tree.
Delete:Splay parent of removed node which is either the node
with the deleted key or its successor.
Cheruku Ravi Teja
Splay Tree
Introduction
Advantages and Disadvantages
Algorithm
TIME COMPLEXITY
Operations
When to Splay
Splaying
Zig step
Zig-Zig step
Zig-Zag step
Results of Splaying
Insertion
Deletion
Splaying
To perform a splay operation , there is need to carry out sequence
of Splay steps, which moves the node closer to the root. The
recently accessed nodes are kept closer to the root so that tree
remains balanced.
Each Splay step depends on three factors
X is left or right child of its parent node P.
Check P is root node or not, if not.
P is left or right child of its parent G.
The three types of Splay steps are
Cheruku Ravi Teja
Splay Tree
Introduction
Advantages and Disadvantages
Algorithm
TIME COMPLEXITY
Operations
When to Splay
Splaying
Zig step
Zig-Zig step
Zig-Zag step
Results of Splaying
Insertion
Deletion
Zig Step
Let X be a non root node on the access path on which we are
rotating.
If the parent P of X is the root of the tree, we merely rotate X
and the root.
This is same as single rotation.
Cheruku Ravi Teja
Splay Tree
Introduction
Advantages and Disadvantages
Algorithm
TIME COMPLEXITY
Operations
When to Splay
Splaying
Zig step
Zig-Zig step
Zig-Zag step
Results of Splaying
Insertion
Deletion
Zig-zig step
Here X and P are either both left children or both right
children.
The Zig-Zig splay rotates between P and G and X and P.
Cheruku Ravi Teja
Splay Tree
Introduction
Advantages and Disadvantages
Algorithm
TIME COMPLEXITY
Operations
When to Splay
Splaying
Zig step
Zig-Zig step
Zig-Zag step
Results of Splaying
Insertion
Deletion
Zig-Zag step
In this case, X and both a parent P and a grandparent G. X is
a right child and P is a left child or vice versa.
This is same as the double rotation.
Cheruku Ravi Teja
Splay Tree
Introduction
Advantages and Disadvantages
Algorithm
TIME COMPLEXITY
Operations
When to Splay
Splaying
Zig step
Zig-Zig step
Zig-Zag step
Results of Splaying
Insertion
Deletion
Results of Splaying
The result is a binary tree, with the left subtree having all
keys less than the root, and the right subtree having keys
greater than the root.
The resulted tree is more balanced than the original tree.
If an operation near the root is done, the tree can become less
balanced.
Cheruku Ravi Teja
Splay Tree
Introduction
Advantages and Disadvantages
Algorithm
TIME COMPLEXITY
Operations
When to Splay
Splaying
Zig step
Zig-Zig step
Zig-Zag step
Results of Splaying
Insertion
Deletion
Insertion
To insert a node in to the tree
Insert a node normally in to the tree.
Splay the newly inserted node to the top of the tree.
Cheruku Ravi Teja
Splay Tree
Introduction
Advantages and Disadvantages
Algorithm
TIME COMPLEXITY
Operations
When to Splay
Splaying
Zig step
Zig-Zig step
Zig-Zag step
Results of Splaying
Insertion
Deletion
Deletion
Access the node to be deleted bringing it to the root.
Delete the root leaving two subtrees L left and R right.
Find the largest element in L, thus the root of L will have no
right child.
Make R the right child of L’s root.
Cheruku Ravi Teja
Splay Tree
Introduction
Advantages and Disadvantages
Algorithm
TIME COMPLEXITY
Operations
When to Splay
Splaying
Zig step
Zig-Zig step
Zig-Zag step
Results of Splaying
Insertion
Deletion
Delete element 6
Cheruku Ravi Teja
Splay Tree
Introduction
Advantages and Disadvantages
Algorithm
TIME COMPLEXITY
TIME COMPLEXITY
For 0 m0 operations to be performed on splay tree it requires
O(mlogn) steps where n is the size of the tree.On an average the
time complexity of each operation on the splay tree is
Search:Searching for a node in the tree would take O(logn).
Insert:Inserting a node in to the tree takes O(logn).
Delete:Deleting a node from the tree takes O(logn).
Cheruku Ravi Teja
Splay Tree
Introduction
Advantages and Disadvantages
Algorithm
TIME COMPLEXITY
References
Data Structures and Their Algorithms, Lewis and Denenberg,
Harper Collins
Algorithms by S Dasgupta U.V Vazirani
wikipedia splaytree http://en.wikipedia.org/wiki/Splay tree
Self-Adjusting Binary Search Trees DANIEL DOMINIC
SLEATOR AND ROBERT ENDRE TARJAN
Cheruku Ravi Teja
Splay Tree